real analysis - A variant of $lim_{n to infty }underbrace{sin sin dotssin(t)}_{text{$n$ compositions}}$

In this question

it's proved that $$\lim_{n \to \infty }\underbrace{\sin \sin \dots\sin(t)}_{\text{$n$ compositions}}, t\in \mathbb {R}$$ converges to the null-function.

In this case the (we can't use Did's argument) $$\lim_{n \to \infty }\underbrace{\sin (2\pi\sin( \dots2\pi\sin(t))}_{\text{$n$ compositions}}$$
what happen?

Answer

Behavior of sequences like these falls into the field of mathematics called dynamical systems. A dynamical system consists of a metric space $X$ and a continuous function $f:X\to X$. Here, $X=\mathbb R$, and $f(x)=2\pi\sin(x)$. One of the questions asked by those studying a dynamical system is what is the behavior of sequences of the form $\{f(x),f(f(x)),f(f(f(x))),...\}$, which is exactly what you are curious about. In this field, we denote the function composed with itself $n$ times by $f^n(x)$. Some dynamical systems are simple and predictable. Others are complicated and chaotic. This is one of the latter types. To demonstrate this, here are some plots:

$\uparrow$ This is $f(x)$.

$\uparrow$ This is $f^2(x)$.

$\uparrow$ This is $f^3(x)$.

$\uparrow$ This is $f^5(x)$.

$\uparrow$ This is $f^8(x)$.

This is chaos$^\ast$! Chaos is, in my opinion, the coolest thing about dynamical systems. Chaos means that even if $x$ and $y$ are close, $f^n(x)$ and $f^n(y)$ may be far apart. Chaos means that it is very difficult to predict how the system will behave in the long term. And chaos means that your sequence certainly doesn't converge to anything.

Some questions that would be interesting to answer are:

• What does the set of all points $x$ such that $0\leq f^n(x)\leq 1$ for all $n$ look like?


• Are there any periodic points, points such that $f^n(x)=x$ for some $n$? (the answer to this is yes). What does the set of periodic points look like?


• If we change $f$ to $f(x)=a\sin(x)$, for $1\leq a\leq 2\pi$, how does the system change? We know if $a=1$, the system is simple, so somewhere, the behavior changes.

$\ast$ It is almost certainly chaotic. We would need a proof to know for sure.